% Copyright 2019 by Mark Wibrow % % This file may be distributed and/or modified % % 1. under the LaTeX Project Public License and/or % 2. under the GNU Public License. % % See the file doc/generic/pgf/licenses/LICENSE for more details. % This file defines the trigonometric functions/functions. % % Much of this file is based on ideas and code (particularly % the old \pgfcoremath.code.tex) due to Till Tantau. % % Version 1.414213 29/9/2007 % \pgfmathpi % \pgfmathdeclarefunction{pi}{0}{\def\pgfmathresult{3.141592654}} \pgfkeys{ /pgf/trig format/.is choice, % attention: the luamath library will hook into these code keys. /pgf/trig format/deg/.code={\def\pgfmath@trig@format@choice{0}}, /pgf/trig format/rad/.code={\def\pgfmath@trig@format@choice{1}}, /pgf/trig format/deg, } % Executes #1 if 'trig format=deg' and #2 otherwise. \def\pgfmathiftrigonometricusesdeg#1#2{% \if0\pgfmath@trig@format@choice #1% \else #2% \fi }% % \pgfmathradians % (for compatibility with 1.18 beta, will be deprecated) % % Convert #1 from radians to degrees (accurate to 1 deg). % \pgfmathdeclarefunction{radians}{1}{% \begingroup% \expandafter\pgfmath@x#1pt\relax% \pgfmath@x57.29577\pgfmath@x\relax% 57.29577 = 360/(2*pi) \pgfmath@returnone\pgfmath@x% \endgroup% }% % \pgfmathdeg % % Convert #1 from radians to degrees. % \pgfmathdeclarefunction{deg}{1}{% \begingroup% \expandafter\pgfmath@x#1pt\relax% \pgfmath@scale@rad@to@deg\pgfmath@x \pgfmath@returnone\pgfmath@x% \endgroup% }% \def\pgfmath@scale@rad@to@deg#1{% #1=57.29577#1\relax% 57.29577 = 360/(2*pi) }% % \pgfmathrad % % Convert #1 from degrees to radians. % \pgfmathdeclarefunction{rad}{1}{% \begingroup% \expandafter\pgfmath@x#1pt\relax% \pgfmath@scale@deg@to@rad\pgfmath@x \pgfmath@returnone\pgfmath@x% \endgroup% }% \def\pgfmath@scale@deg@to@rad#1{% #1=3.14159#1% \divide#1180\relax% }% % \pgfmathsin % % Calculate the sine of #1 (in degrees). % \pgfmathdeclarefunction{sin}{1}{% % Let #1 = a.b % Then sin(#1) \approx (1-b)*sin(a) + b*sin(a+1) % \begingroup% \expandafter\pgfmath@x#1pt\relax% \if1\pgfmath@trig@format@choice % trig format=rad \pgfmath@scale@rad@to@deg\pgfmath@x \fi \advance\pgfmath@x-90pt\relax% \afterassignment\pgfmath@gobbletilpgfmath@ \expandafter\c@pgfmath@counta\the\pgfmath@x\relax\pgfmath@ \divide\c@pgfmath@counta360\relax% \multiply\c@pgfmath@counta-360\relax% \advance\pgfmath@x\c@pgfmath@counta pt\relax% \ifdim\pgfmath@x<0pt\relax\pgfmath@x-\pgfmath@x\fi% \ifdim\pgfmath@x<180pt\relax% \else \pgfmath@x-\pgfmath@x% \advance\pgfmath@x360pt\relax% \fi% % Now 0 <= x < 179. So split x into integer and decimal... % \afterassignment\pgfmath@xa% \expandafter\c@pgfmath@counta\the\pgfmath@x\relax% % % ...if #1 is an integer, don't do anything fancy. \ifdim\pgfmath@xa=0pt% \expandafter\pgfmath@x\csname pgfmath@cos@\the\c@pgfmath@counta\endcsname pt\relax% \else% \pgfmath@x-\pgfmath@xa% \advance\pgfmath@x1pt\relax% \expandafter\pgfmath@x\csname pgfmath@cos@\the\c@pgfmath@counta\endcsname\pgfmath@x% \advance\c@pgfmath@counta1\relax% \ifnum\c@pgfmath@counta=181\relax\c@pgfmath@counta179\relax\fi% \expandafter\advance\expandafter\pgfmath@x\csname pgfmath@cos@\the\c@pgfmath@counta\endcsname\pgfmath@xa% \fi% \pgfmath@returnone\pgfmath@x% \endgroup } % \pgfmathcos % % Calculate the cosine of #1 (in degrees). % \pgfmathdeclarefunction{cos}{1}{% % Let x = a.b % Then cos(x) \approx (1-b)*cos(a) + b*cos(a+1) % \begingroup% \expandafter\pgfmath@x#1pt\relax% \if1\pgfmath@trig@format@choice % trig format=rad \pgfmath@scale@rad@to@deg\pgfmath@x \fi \afterassignment\pgfmath@gobbletilpgfmath@% \expandafter\c@pgfmath@counta\the\pgfmath@x\relax\pgfmath@% \divide\c@pgfmath@counta360\relax% \multiply\c@pgfmath@counta-360\relax% \advance\pgfmath@x\c@pgfmath@counta pt\relax% \ifdim\pgfmath@x<0pt\relax\pgfmath@x-\pgfmath@x\fi% \ifdim\pgfmath@x<180pt\relax% \else% \pgfmath@x-\pgfmath@x% \advance\pgfmath@x360pt\relax% \fi% % Now 0 <= x < 179. So split x into integer and decimal... % \afterassignment\pgfmath@xa% \expandafter\c@pgfmath@counta\the\pgfmath@x\relax% % % ...if #1 is an integer, don't do anything fancy. \ifdim\pgfmath@xa=0pt% \expandafter\pgfmath@x\csname pgfmath@cos@\the\c@pgfmath@counta\endcsname pt% \else% \pgfmath@x-\pgfmath@xa% \advance\pgfmath@x1pt\relax% \expandafter\pgfmath@x\csname pgfmath@cos@\the\c@pgfmath@counta\endcsname\pgfmath@x% \advance\c@pgfmath@counta1\relax% \ifnum\c@pgfmath@counta=181\relax\c@pgfmath@counta179\relax\fi% \expandafter\advance\expandafter\pgfmath@x\csname pgfmath@cos@\the\c@pgfmath@counta\endcsname\pgfmath@xa% \fi% \pgfmath@returnone\pgfmath@x% \endgroup% } % \pgfmathsincos % % Calculate the sin and cosine of #1 (in degrees). % \def\pgfmathsincos#1{% \pgfmathparse{#1}% \expandafter\pgfmathsincos@\expandafter{\pgfmathresult}} \def\pgfmathsincos@#1{% \edef\pgfmath@temparg{#1}% \pgfmathsin@{\pgfmath@temparg}\edef\pgfmathresulty{\pgfmathresult}% \pgfmathcos@{\pgfmath@temparg}\edef\pgfmathresultx{\pgfmathresult}% } % \pgfmathtan % % Calculate the tangent of #1 (in degrees). % \pgfmathdeclarefunction{tan}{1}{% \begingroup% \edef\pgfmath@temparg{#1}% \pgfmathcos@{\pgfmath@temparg}% \expandafter\pgfmathreciprocal@\expandafter{\pgfmathresult}% \edef\pgfmath@tantemp{\pgfmathresult}% \pgfmathsin@{\pgfmath@temparg}% \pgfmath@x\pgfmathresult pt\relax% \pgfmath@x\pgfmath@tantemp\pgfmath@x% \pgfmath@returnone\pgfmath@x% \endgroup% } % \pgfmathcosec % % Calculate the cosec of #1 (in degrees). % \pgfmathdeclarefunction{cosec}{1}{% \expandafter\pgfmathsin@\expandafter{#1}% \pgfmathreciprocal@{\pgfmathresult}% } % \pgfmathsec % % Calculate the sec of #1 (in degrees). % \pgfmathdeclarefunction{sec}{1}{% \expandafter\pgfmathcos@\expandafter{#1}% \pgfmathreciprocal@{\pgfmathresult}% } % \pgfmathcot % % Calculate the cotangent of #1 (in degrees). % \pgfmathdeclarefunction{cot}{1}{% \begingroup% \edef\pgfmath@temparg{#1}% \pgfmathsin@{\pgfmath@temparg}% \expandafter\pgfmathreciprocal@\expandafter{\pgfmathresult}% \edef\pgfmath@tantemp{\pgfmathresult}% \pgfmathcos@{\pgfmath@temparg}% \pgfmath@x\pgfmathresult pt\relax% \pgfmath@x\pgfmath@tantemp\pgfmath@x% \pgfmath@returnone\pgfmath@x% \endgroup% } \def\pgfmath@multiply@thousand#1.#2#3#4#5\relax{% \edef\pgfmathresult{#1#2#3#4.#5}% }% % \pgfmathasin % % The asin of #1 % \pgfmathdeclarefunction{asin}{1}{% \begingroup% \pgfmath@x=#1pt % \pgfmath@xa\pgfmath@x% \edef\pgf@temp{\pgfmath@tonumber\pgfmath@x}% % compute lossless '*1000' operation: \expandafter\pgfmath@multiply@thousand\pgf@temp 000\relax \pgfmath@x=\pgfmathresult pt % \ifdim\pgfmath@x<0pt\relax% \pgfmath@x-\pgfmath@x% \fi% \pgfmath@table@lookup{\pgfmath@x}{pgfmath@acos@}{1001}% \pgfmath@x=-\pgfmath@x \advance\pgfmath@x90pt\relax% \ifdim\pgfmath@xa<0pt\relax% \pgfmath@x-\pgfmath@x% \fi% \if1\pgfmath@trig@format@choice % trig format=rad \pgfmath@scale@deg@to@rad\pgfmath@x \fi \pgfmath@returnone\pgfmath@x% \endgroup% } % Defines \pgfmath@x to be the result of the table lookup for value % '#1' in table '#2' % % #1: a value for which lookup should be performed % #2: the table's prefix name % #3: the table's size \def\pgfmath@table@lookup#1#2#3{% \pgfmath@x=#1\relax \expandafter\pgfmath@table@lookup@\the\pgfmath@x\relax{#2}{#3}% } \def\pgfmath@table@lookup@#1.#2\relax#3#4{% \edef\pgf@temp{\Pgf@geT#2}% strip 'pt' suffix. % % prepare constant interpolant (i.e. the value at node '#1'): \pgfmath@x=\csname #3#1\endcsname pt % % \c@pgfmath@countc=#1 % \advance\c@pgfmath@countc by1 % \ifnum\c@pgfmath@countc<#4 % % Ah - we have enough elements in the table. % % perform linear interpolation: \pgfmath@ya=\csname #3\the\c@pgfmath@countc\endcsname pt % \advance\pgfmath@ya by-\pgfmath@x % \advance\pgfmath@x by0.\pgf@temp\pgfmath@ya \fi }% % \pgfmathacos % % The acos of #1 % \pgfmathdeclarefunction{acos}{1}{% \begingroup% \pgfmath@x#1pt\relax% \pgfmath@xa\pgfmath@x% \edef\pgf@temp{\pgfmath@tonumber\pgfmath@x}% % compute lossless '*1000' operation: \expandafter\pgfmath@multiply@thousand\pgf@temp 000\relax \pgfmath@x=\pgfmathresult pt % \ifdim\pgfmath@x<0pt\relax% \pgfmath@x-\pgfmath@x% \fi% \pgfmath@table@lookup{\pgfmath@x}{pgfmath@acos@}{1001}% \ifdim\pgfmath@xa<0pt\relax% \pgfmath@x-\pgfmath@x% \advance\pgfmath@x by180pt\relax% \fi% \if1\pgfmath@trig@format@choice % trig format=rad \pgfmath@scale@deg@to@rad\pgfmath@x \fi \pgfmath@returnone\pgfmath@x% \endgroup% } \pgfmathdeclarefunction{atan}{1}{% \begingroup% \expandafter\pgfmath@x#1pt\relax% \pgfmath@xa\pgfmath@x% \ifdim\pgfmath@x<0pt\relax% \pgfmath@x-\pgfmath@x% \fi% \pgfmath@xb\pgfmath@x% \ifdim\pgfmath@x>1pt\relax% \edef\pgfmath@temp{\pgfmath@tonumber{\pgfmath@x}}% \pgfmathreciprocal@{\pgfmath@temp}% \pgfmath@x\pgfmathresult pt\relax% \fi% % compute lossless '*1000' operation: \edef\pgf@temp{\pgfmath@tonumber\pgfmath@x}% \expandafter\pgfmath@multiply@thousand\pgf@temp 000\relax \pgfmath@x=\pgfmathresult pt % \pgfmath@table@lookup{\pgfmath@x}{pgfmath@atan@}{1001}%xxx \ifdim\pgfmath@xb>1pt\relax% \pgfmath@x=-\pgfmath@x% \advance\pgfmath@x90pt% \fi% \ifdim\pgfmath@xa<0pt\relax% \pgfmath@x-\pgfmath@x% \fi% \if1\pgfmath@trig@format@choice % trig format=rad \pgfmath@scale@deg@to@rad\pgfmath@x \fi \pgfmath@returnone\pgfmath@x% \endgroup% } \pgfmathdeclarefunction{atan2}{2}{% % Note: first parameter is y (!), second is x (!) \begingroup% \let\pgfmath@trig@format@choice@@=\pgfmath@trig@format@choice \def\pgfmath@trig@format@choice{0}% \pgfmath@y=#1pt\relax% \ifdim\pgfmath@y<0pt\relax% \pgfmath@y=-\pgfmath@y% \fi% \ifdim\pgfmath@y<0.001pt\relax% \ifdim#2pt<0pt\relax% \pgfmath@x=+180pt\relax% \else% \ifdim#2pt>0pt\relax% \pgfmath@x=0pt\relax% \else% \ifdim#1pt<0pt% \pgfmath@x=-90pt\relax% \else% \pgfmath@x=90pt\relax% \fi% \fi% \fi% \else% \pgfmathabs@{#1}\let\pgfmath@tempa\pgfmathresult% \pgfmathabs@{#2}\let\pgfmath@tempb\pgfmathresult% \ifdim\pgfmath@tempa pt>\pgfmath@tempb pt% \pgfmathdivide@{#2}{\pgfmath@tempa}% \expandafter\pgfmathatan@\expandafter{\pgfmathresult}% \pgfmath@x=90pt% \advance\pgfmath@x by-\pgfmathresult pt\relax% \else% \pgfmathdivide@{\pgfmath@tempa}{#2}% \expandafter\pgfmathatan@\expandafter{\pgfmathresult}% \pgfmath@x=\pgfmathresult pt\relax% \ifdim#2pt<0pt% \advance\pgfmath@x by180pt\relax% \fi% \fi% \ifdim#1pt<0pt\relax% \pgfmath@x=-\pgfmath@x% \fi% \fi% \if1\pgfmath@trig@format@choice@@ % trig format=rad \pgfmath@scale@deg@to@rad\pgfmath@x \fi \pgfmath@returnone\pgfmath@x% \endgroup% } % This is needed since \pgfmathatan2 is not a legal macro name \expandafter\let\expandafter\pgfmathatantwo\csname pgfmathatan2\endcsname \expandafter\let\expandafter\pgfmathatantwo@\csname pgfmathatan2@\endcsname \def\pgfmath@def#1#2#3{\expandafter\def\csname pgfmath@#1@#2\endcsname{#3}} \pgfmath@def{cos}{0}{1.00000} \pgfmath@def{cos}{1}{0.99985} \pgfmath@def{cos}{2}{0.99939} \pgfmath@def{cos}{3}{0.99863} \pgfmath@def{cos}{4}{0.99756} \pgfmath@def{cos}{5}{0.99619} \pgfmath@def{cos}{6}{0.99452} \pgfmath@def{cos}{7}{0.99255} \pgfmath@def{cos}{8}{0.99027} \pgfmath@def{cos}{9}{0.98769} \pgfmath@def{cos}{10}{0.98481} \pgfmath@def{cos}{11}{0.98163} \pgfmath@def{cos}{12}{0.97815} \pgfmath@def{cos}{13}{0.97437} \pgfmath@def{cos}{14}{0.97030} \pgfmath@def{cos}{15}{0.96593} \pgfmath@def{cos}{16}{0.96126} \pgfmath@def{cos}{17}{0.95630} \pgfmath@def{cos}{18}{0.95106} \pgfmath@def{cos}{19}{0.94552} \pgfmath@def{cos}{20}{0.93969} \pgfmath@def{cos}{21}{0.93358} \pgfmath@def{cos}{22}{0.92718} \pgfmath@def{cos}{23}{0.92050} \pgfmath@def{cos}{24}{0.91355} \pgfmath@def{cos}{25}{0.90631} \pgfmath@def{cos}{26}{0.89879} \pgfmath@def{cos}{27}{0.89101} \pgfmath@def{cos}{28}{0.88295} \pgfmath@def{cos}{29}{0.87462} \pgfmath@def{cos}{30}{0.86603} \pgfmath@def{cos}{31}{0.85717} \pgfmath@def{cos}{32}{0.84805} \pgfmath@def{cos}{33}{0.83867} \pgfmath@def{cos}{34}{0.82904} \pgfmath@def{cos}{35}{0.81915} \pgfmath@def{cos}{36}{0.80902} \pgfmath@def{cos}{37}{0.79864} \pgfmath@def{cos}{38}{0.78801} \pgfmath@def{cos}{39}{0.77715} \pgfmath@def{cos}{40}{0.76604} \pgfmath@def{cos}{41}{0.75471} \pgfmath@def{cos}{42}{0.74314} \pgfmath@def{cos}{43}{0.73135} \pgfmath@def{cos}{44}{0.71934} \pgfmath@def{cos}{45}{0.70711} \pgfmath@def{cos}{46}{0.69466} \pgfmath@def{cos}{47}{0.68200} \pgfmath@def{cos}{48}{0.66913} \pgfmath@def{cos}{49}{0.65606} \pgfmath@def{cos}{50}{0.64279} \pgfmath@def{cos}{51}{0.62932} \pgfmath@def{cos}{52}{0.61566} \pgfmath@def{cos}{53}{0.60182} \pgfmath@def{cos}{54}{0.58779} \pgfmath@def{cos}{55}{0.57358} \pgfmath@def{cos}{56}{0.55919} \pgfmath@def{cos}{57}{0.54464} \pgfmath@def{cos}{58}{0.52992} \pgfmath@def{cos}{59}{0.51504} \pgfmath@def{cos}{60}{0.50000} \pgfmath@def{cos}{61}{0.48481} \pgfmath@def{cos}{62}{0.46947} \pgfmath@def{cos}{63}{0.45399} \pgfmath@def{cos}{64}{0.43837} \pgfmath@def{cos}{65}{0.42262} \pgfmath@def{cos}{66}{0.40674} \pgfmath@def{cos}{67}{0.39073} \pgfmath@def{cos}{68}{0.37461} \pgfmath@def{cos}{69}{0.35837} \pgfmath@def{cos}{70}{0.34202} \pgfmath@def{cos}{71}{0.32557} \pgfmath@def{cos}{72}{0.30902} \pgfmath@def{cos}{73}{0.29237} \pgfmath@def{cos}{74}{0.27564} \pgfmath@def{cos}{75}{0.25882} \pgfmath@def{cos}{76}{0.24192} \pgfmath@def{cos}{77}{0.22495} \pgfmath@def{cos}{78}{0.20791} \pgfmath@def{cos}{79}{0.19081} \pgfmath@def{cos}{80}{0.17365} \pgfmath@def{cos}{81}{0.15643} \pgfmath@def{cos}{82}{0.13917} \pgfmath@def{cos}{83}{0.12187} \pgfmath@def{cos}{84}{0.10453} \pgfmath@def{cos}{85}{0.08716} \pgfmath@def{cos}{86}{0.06976} \pgfmath@def{cos}{87}{0.05234} \pgfmath@def{cos}{88}{0.03490} \pgfmath@def{cos}{89}{0.01745} \pgfmath@def{cos}{90}{0.00000} \pgfmath@def{cos}{91}{-0.01745} \pgfmath@def{cos}{92}{-0.03490} \pgfmath@def{cos}{93}{-0.05234} \pgfmath@def{cos}{94}{-0.06976} \pgfmath@def{cos}{95}{-0.08716} \pgfmath@def{cos}{96}{-0.10453} \pgfmath@def{cos}{97}{-0.12187} \pgfmath@def{cos}{98}{-0.13917} \pgfmath@def{cos}{99}{-0.15643} \pgfmath@def{cos}{100}{-0.17365} \pgfmath@def{cos}{101}{-0.19081} \pgfmath@def{cos}{102}{-0.20791} \pgfmath@def{cos}{103}{-0.22495} \pgfmath@def{cos}{104}{-0.24192} \pgfmath@def{cos}{105}{-0.25882} \pgfmath@def{cos}{106}{-0.27564} \pgfmath@def{cos}{107}{-0.29237} \pgfmath@def{cos}{108}{-0.30902} \pgfmath@def{cos}{109}{-0.32557} \pgfmath@def{cos}{110}{-0.34202} \pgfmath@def{cos}{111}{-0.35837} \pgfmath@def{cos}{112}{-0.37461} \pgfmath@def{cos}{113}{-0.39073} \pgfmath@def{cos}{114}{-0.40674} \pgfmath@def{cos}{115}{-0.42262} \pgfmath@def{cos}{116}{-0.43837} \pgfmath@def{cos}{117}{-0.45399} \pgfmath@def{cos}{118}{-0.46947} \pgfmath@def{cos}{119}{-0.48481} \pgfmath@def{cos}{120}{-0.50000} \pgfmath@def{cos}{121}{-0.51504} \pgfmath@def{cos}{122}{-0.52992} \pgfmath@def{cos}{123}{-0.54464} \pgfmath@def{cos}{124}{-0.55919} \pgfmath@def{cos}{125}{-0.57358} \pgfmath@def{cos}{126}{-0.58779} \pgfmath@def{cos}{127}{-0.60182} \pgfmath@def{cos}{128}{-0.61566} \pgfmath@def{cos}{129}{-0.62932} \pgfmath@def{cos}{130}{-0.64279} \pgfmath@def{cos}{131}{-0.65606} \pgfmath@def{cos}{132}{-0.66913} \pgfmath@def{cos}{133}{-0.68200} \pgfmath@def{cos}{134}{-0.69466} \pgfmath@def{cos}{135}{-0.70711} \pgfmath@def{cos}{136}{-0.71934} \pgfmath@def{cos}{137}{-0.73135} \pgfmath@def{cos}{138}{-0.74314} \pgfmath@def{cos}{139}{-0.75471} \pgfmath@def{cos}{140}{-0.76604} \pgfmath@def{cos}{141}{-0.77715} \pgfmath@def{cos}{142}{-0.78801} \pgfmath@def{cos}{143}{-0.79864} \pgfmath@def{cos}{144}{-0.80902} \pgfmath@def{cos}{145}{-0.81915} \pgfmath@def{cos}{146}{-0.82904} \pgfmath@def{cos}{147}{-0.83867} \pgfmath@def{cos}{148}{-0.84805} \pgfmath@def{cos}{149}{-0.85717} \pgfmath@def{cos}{150}{-0.86603} \pgfmath@def{cos}{151}{-0.87462} \pgfmath@def{cos}{152}{-0.88295} \pgfmath@def{cos}{153}{-0.89101} \pgfmath@def{cos}{154}{-0.89879} \pgfmath@def{cos}{155}{-0.90631} \pgfmath@def{cos}{156}{-0.91355} \pgfmath@def{cos}{157}{-0.92050} \pgfmath@def{cos}{158}{-0.92718} \pgfmath@def{cos}{159}{-0.93358} \pgfmath@def{cos}{160}{-0.93969} \pgfmath@def{cos}{161}{-0.94552} \pgfmath@def{cos}{162}{-0.95106} \pgfmath@def{cos}{163}{-0.95630} \pgfmath@def{cos}{164}{-0.96126} \pgfmath@def{cos}{165}{-0.96593} \pgfmath@def{cos}{166}{-0.97030} \pgfmath@def{cos}{167}{-0.97437} \pgfmath@def{cos}{168}{-0.97815} \pgfmath@def{cos}{169}{-0.98163} \pgfmath@def{cos}{170}{-0.98481} \pgfmath@def{cos}{171}{-0.98769} \pgfmath@def{cos}{172}{-0.99027} \pgfmath@def{cos}{173}{-0.99255} \pgfmath@def{cos}{174}{-0.99452} \pgfmath@def{cos}{175}{-0.99619} \pgfmath@def{cos}{176}{-0.99756} \pgfmath@def{cos}{177}{-0.99863} \pgfmath@def{cos}{178}{-0.99939} \pgfmath@def{cos}{179}{-0.99985} \pgfmath@def{cos}{180}{-1.00000} \pgfmath@def{cos}{181}{-0.99985} \pgfmath@def{cosfrac}{0}{1.00000} \pgfmath@def{cosfrac}{1}{0.99995} \pgfmath@def{cosfrac}{2}{0.99980} \pgfmath@def{cosfrac}{3}{0.99955} \pgfmath@def{cosfrac}{4}{0.99920} \pgfmath@def{cosfrac}{5}{0.99875} \pgfmath@def{cosfrac}{6}{0.99820} \pgfmath@def{cosfrac}{7}{0.99755} \pgfmath@def{cosfrac}{8}{0.99681} \pgfmath@def{cosfrac}{9}{0.99597} \pgfmath@def{cosfrac}{10}{0.99503} \pgfmath@def{cosfrac}{11}{0.99400} \pgfmath@def{cosfrac}{12}{0.99287} \pgfmath@def{cosfrac}{13}{0.99165} \pgfmath@def{cosfrac}{14}{0.99034} \pgfmath@def{cosfrac}{15}{0.98893} \pgfmath@def{cosfrac}{16}{0.98744} \pgfmath@def{cosfrac}{17}{0.98585} \pgfmath@def{cosfrac}{18}{0.98418} \pgfmath@def{cosfrac}{19}{0.98242} \pgfmath@def{cosfrac}{20}{0.98058} \pgfmath@def{cosfrac}{21}{0.97865} \pgfmath@def{cosfrac}{22}{0.97664} \pgfmath@def{cosfrac}{23}{0.97455} \pgfmath@def{cosfrac}{24}{0.97238} \pgfmath@def{cosfrac}{25}{0.97014} \pgfmath@def{cosfrac}{26}{0.96782} \pgfmath@def{cosfrac}{27}{0.96542} \pgfmath@def{cosfrac}{28}{0.96296} \pgfmath@def{cosfrac}{29}{0.96042} \pgfmath@def{cosfrac}{30}{0.95782} \pgfmath@def{cosfrac}{31}{0.95515} \pgfmath@def{cosfrac}{32}{0.95242} \pgfmath@def{cosfrac}{33}{0.94962} \pgfmath@def{cosfrac}{34}{0.94677} \pgfmath@def{cosfrac}{35}{0.94385} \pgfmath@def{cosfrac}{36}{0.94088} \pgfmath@def{cosfrac}{37}{0.93786} \pgfmath@def{cosfrac}{38}{0.93478} \pgfmath@def{cosfrac}{39}{0.93165} \pgfmath@def{cosfrac}{40}{0.92847} \pgfmath@def{cosfrac}{41}{0.92525} \pgfmath@def{cosfrac}{42}{0.92198} \pgfmath@def{cosfrac}{43}{0.91866} \pgfmath@def{cosfrac}{44}{0.91531} \pgfmath@def{cosfrac}{45}{0.91192} \pgfmath@def{cosfrac}{46}{0.90849} \pgfmath@def{cosfrac}{47}{0.90502} \pgfmath@def{cosfrac}{48}{0.90152} \pgfmath@def{cosfrac}{49}{0.89799} \pgfmath@def{cosfrac}{50}{0.89442} \pgfmath@def{cosfrac}{51}{0.89083} \pgfmath@def{cosfrac}{52}{0.88721} \pgfmath@def{cosfrac}{53}{0.88357} \pgfmath@def{cosfrac}{54}{0.87990} \pgfmath@def{cosfrac}{55}{0.87621} \pgfmath@def{cosfrac}{56}{0.87250} \pgfmath@def{cosfrac}{57}{0.86877} \pgfmath@def{cosfrac}{58}{0.86503} \pgfmath@def{cosfrac}{59}{0.86126} \pgfmath@def{cosfrac}{60}{0.85749} \pgfmath@def{cosfrac}{61}{0.85370} \pgfmath@def{cosfrac}{62}{0.84990} \pgfmath@def{cosfrac}{63}{0.84609} \pgfmath@def{cosfrac}{64}{0.84227} \pgfmath@def{cosfrac}{65}{0.83844} \pgfmath@def{cosfrac}{66}{0.83460} \pgfmath@def{cosfrac}{67}{0.83077} \pgfmath@def{cosfrac}{68}{0.82692} \pgfmath@def{cosfrac}{69}{0.82308} \pgfmath@def{cosfrac}{70}{0.81923} \pgfmath@def{cosfrac}{71}{0.81538} \pgfmath@def{cosfrac}{72}{0.81153} \pgfmath@def{cosfrac}{73}{0.80768} \pgfmath@def{cosfrac}{74}{0.80384} \pgfmath@def{cosfrac}{75}{0.80000} \pgfmath@def{cosfrac}{76}{0.79616} \pgfmath@def{cosfrac}{77}{0.79232} \pgfmath@def{cosfrac}{78}{0.78850} \pgfmath@def{cosfrac}{79}{0.78468} \pgfmath@def{cosfrac}{80}{0.78086} \pgfmath@def{cosfrac}{81}{0.77706} \pgfmath@def{cosfrac}{82}{0.77326} \pgfmath@def{cosfrac}{83}{0.76948} \pgfmath@def{cosfrac}{84}{0.76570} \pgfmath@def{cosfrac}{85}{0.76193} \pgfmath@def{cosfrac}{86}{0.75818} \pgfmath@def{cosfrac}{87}{0.75444} \pgfmath@def{cosfrac}{88}{0.75071} \pgfmath@def{cosfrac}{89}{0.74699} \pgfmath@def{cosfrac}{90}{0.74329} \pgfmath@def{cosfrac}{91}{0.73960} \pgfmath@def{cosfrac}{92}{0.73593} \pgfmath@def{cosfrac}{93}{0.73227} \pgfmath@def{cosfrac}{94}{0.72862} \pgfmath@def{cosfrac}{95}{0.72499} \pgfmath@def{cosfrac}{96}{0.72138} \pgfmath@def{cosfrac}{97}{0.71779} \pgfmath@def{cosfrac}{98}{0.71421} \pgfmath@def{cosfrac}{99}{0.71065} \pgfmath@def{cosfrac}{100}{0.70710} % Oh No! Some really loooooong tables. \pgfmath@def{atan}{0}{0.00000} \pgfmath@def{atan}{1}{0.05729} \pgfmath@def{atan}{2}{0.11459} \pgfmath@def{atan}{3}{0.17188} \pgfmath@def{atan}{4}{0.22918} \pgfmath@def{atan}{5}{0.28647} \pgfmath@def{atan}{6}{0.34377} \pgfmath@def{atan}{7}{0.40106} \pgfmath@def{atan}{8}{0.45835} \pgfmath@def{atan}{9}{0.51564} \pgfmath@def{atan}{10}{0.57293} \pgfmath@def{atan}{11}{0.63022} \pgfmath@def{atan}{12}{0.68751} \pgfmath@def{atan}{13}{0.74480} \pgfmath@def{atan}{14}{0.80208} \pgfmath@def{atan}{15}{0.85937} \pgfmath@def{atan}{16}{0.91665} \pgfmath@def{atan}{17}{0.97393} \pgfmath@def{atan}{18}{1.03121} \pgfmath@def{atan}{19}{1.08848} \pgfmath@def{atan}{20}{1.14576} \pgfmath@def{atan}{21}{1.20303} \pgfmath@def{atan}{22}{1.26030} \pgfmath@def{atan}{23}{1.31757} \pgfmath@def{atan}{24}{1.37483} \pgfmath@def{atan}{25}{1.43209} \pgfmath@def{atan}{26}{1.48935} \pgfmath@def{atan}{27}{1.54661} \pgfmath@def{atan}{28}{1.60386} \pgfmath@def{atan}{29}{1.66111} \pgfmath@def{atan}{30}{1.71835} \pgfmath@def{atan}{31}{1.77560} \pgfmath@def{atan}{32}{1.83284} \pgfmath@def{atan}{33}{1.89007} \pgfmath@def{atan}{34}{1.94730} \pgfmath@def{atan}{35}{2.00453} \pgfmath@def{atan}{36}{2.06175} \pgfmath@def{atan}{37}{2.11897} \pgfmath@def{atan}{38}{2.17619} \pgfmath@def{atan}{39}{2.23340} \pgfmath@def{atan}{40}{2.29061} \pgfmath@def{atan}{41}{2.34781} \pgfmath@def{atan}{42}{2.40500} \pgfmath@def{atan}{43}{2.46220} \pgfmath@def{atan}{44}{2.51938} \pgfmath@def{atan}{45}{2.57657} \pgfmath@def{atan}{46}{2.63374} \pgfmath@def{atan}{47}{2.69092} \pgfmath@def{atan}{48}{2.74808} \pgfmath@def{atan}{49}{2.80524} \pgfmath@def{atan}{50}{2.86240} \pgfmath@def{atan}{51}{2.91955} \pgfmath@def{atan}{52}{2.97669} \pgfmath@def{atan}{53}{3.03383} \pgfmath@def{atan}{54}{3.09097} \pgfmath@def{atan}{55}{3.14809} \pgfmath@def{atan}{56}{3.20521} \pgfmath@def{atan}{57}{3.26232} \pgfmath@def{atan}{58}{3.31943} \pgfmath@def{atan}{59}{3.37653} \pgfmath@def{atan}{60}{3.43363} \pgfmath@def{atan}{61}{3.49071} \pgfmath@def{atan}{62}{3.54779} \pgfmath@def{atan}{63}{3.60487} \pgfmath@def{atan}{64}{3.66193} \pgfmath@def{atan}{65}{3.71899} \pgfmath@def{atan}{66}{3.77604} \pgfmath@def{atan}{67}{3.83308} \pgfmath@def{atan}{68}{3.89012} \pgfmath@def{atan}{69}{3.94715} \pgfmath@def{atan}{70}{4.00417} \pgfmath@def{atan}{71}{4.06118} \pgfmath@def{atan}{72}{4.11819} \pgfmath@def{atan}{73}{4.17518} \pgfmath@def{atan}{74}{4.23217} \pgfmath@def{atan}{75}{4.28915} \pgfmath@def{atan}{76}{4.34612} \pgfmath@def{atan}{77}{4.40308} \pgfmath@def{atan}{78}{4.46004} \pgfmath@def{atan}{79}{4.51698} \pgfmath@def{atan}{80}{4.57392} \pgfmath@def{atan}{81}{4.63084} \pgfmath@def{atan}{82}{4.68776} \pgfmath@def{atan}{83}{4.74467} \pgfmath@def{atan}{84}{4.80157} \pgfmath@def{atan}{85}{4.85846} \pgfmath@def{atan}{86}{4.91534} \pgfmath@def{atan}{87}{4.97221} \pgfmath@def{atan}{88}{5.02907} \pgfmath@def{atan}{89}{5.08592} \pgfmath@def{atan}{90}{5.14276} \pgfmath@def{atan}{91}{5.19959} \pgfmath@def{atan}{92}{5.25641} \pgfmath@def{atan}{93}{5.31322} \pgfmath@def{atan}{94}{5.37002} \pgfmath@def{atan}{95}{5.42681} \pgfmath@def{atan}{96}{5.48359} \pgfmath@def{atan}{97}{5.54035} \pgfmath@def{atan}{98}{5.59711} \pgfmath@def{atan}{99}{5.65385} \pgfmath@def{atan}{100}{5.71059} \pgfmath@def{atan}{101}{5.76731} \pgfmath@def{atan}{102}{5.82402} \pgfmath@def{atan}{103}{5.88072} \pgfmath@def{atan}{104}{5.93741} \pgfmath@def{atan}{105}{5.99409} \pgfmath@def{atan}{106}{6.05075} \pgfmath@def{atan}{107}{6.10741} \pgfmath@def{atan}{108}{6.16405} \pgfmath@def{atan}{109}{6.22068} \pgfmath@def{atan}{110}{6.27729} \pgfmath@def{atan}{111}{6.33390} \pgfmath@def{atan}{112}{6.39049} \pgfmath@def{atan}{113}{6.44707} \pgfmath@def{atan}{114}{6.50364} \pgfmath@def{atan}{115}{6.56019} \pgfmath@def{atan}{116}{6.61673} \pgfmath@def{atan}{117}{6.67326} \pgfmath@def{atan}{118}{6.72978} \pgfmath@def{atan}{119}{6.78628} \pgfmath@def{atan}{120}{6.84277} \pgfmath@def{atan}{121}{6.89924} \pgfmath@def{atan}{122}{6.95571} \pgfmath@def{atan}{123}{7.01216} \pgfmath@def{atan}{124}{7.06859} \pgfmath@def{atan}{125}{7.12501} \pgfmath@def{atan}{126}{7.18142} \pgfmath@def{atan}{127}{7.23781} \pgfmath@def{atan}{128}{7.29419} \pgfmath@def{atan}{129}{7.35056} \pgfmath@def{atan}{130}{7.40691} \pgfmath@def{atan}{131}{7.46324} \pgfmath@def{atan}{132}{7.51957} \pgfmath@def{atan}{133}{7.57587} \pgfmath@def{atan}{134}{7.63217} \pgfmath@def{atan}{135}{7.68844} \pgfmath@def{atan}{136}{7.74471} \pgfmath@def{atan}{137}{7.80095} \pgfmath@def{atan}{138}{7.85719} \pgfmath@def{atan}{139}{7.91340} \pgfmath@def{atan}{140}{7.96961} \pgfmath@def{atan}{141}{8.02579} \pgfmath@def{atan}{142}{8.08196} \pgfmath@def{atan}{143}{8.13812} \pgfmath@def{atan}{144}{8.19426} \pgfmath@def{atan}{145}{8.25038} \pgfmath@def{atan}{146}{8.30649} \pgfmath@def{atan}{147}{8.36258} \pgfmath@def{atan}{148}{8.41866} \pgfmath@def{atan}{149}{8.47472} \pgfmath@def{atan}{150}{8.53076} \pgfmath@def{atan}{151}{8.58679} \pgfmath@def{atan}{152}{8.64280} \pgfmath@def{atan}{153}{8.69879} \pgfmath@def{atan}{154}{8.75477} \pgfmath@def{atan}{155}{8.81073} \pgfmath@def{atan}{156}{8.86667} \pgfmath@def{atan}{157}{8.92260} \pgfmath@def{atan}{158}{8.97851} \pgfmath@def{atan}{159}{9.03440} \pgfmath@def{atan}{160}{9.09027} \pgfmath@def{atan}{161}{9.14613} \pgfmath@def{atan}{162}{9.20197} \pgfmath@def{atan}{163}{9.25779} \pgfmath@def{atan}{164}{9.31359} \pgfmath@def{atan}{165}{9.36938} \pgfmath@def{atan}{166}{9.42515} \pgfmath@def{atan}{167}{9.48090} \pgfmath@def{atan}{168}{9.53663} \pgfmath@def{atan}{169}{9.59235} \pgfmath@def{atan}{170}{9.64804} \pgfmath@def{atan}{171}{9.70372} \pgfmath@def{atan}{172}{9.75938} \pgfmath@def{atan}{173}{9.81502} \pgfmath@def{atan}{174}{9.87064} \pgfmath@def{atan}{175}{9.92624} \pgfmath@def{atan}{176}{9.98182} \pgfmath@def{atan}{177}{10.03739} \pgfmath@def{atan}{178}{10.09294} \pgfmath@def{atan}{179}{10.14846} \pgfmath@def{atan}{180}{10.20397} \pgfmath@def{atan}{181}{10.25946} \pgfmath@def{atan}{182}{10.31493} \pgfmath@def{atan}{183}{10.37037} \pgfmath@def{atan}{184}{10.42580} \pgfmath@def{atan}{185}{10.48121} \pgfmath@def{atan}{186}{10.53660} \pgfmath@def{atan}{187}{10.59197} \pgfmath@def{atan}{188}{10.64732} \pgfmath@def{atan}{189}{10.70265} \pgfmath@def{atan}{190}{10.75796} \pgfmath@def{atan}{191}{10.81325} \pgfmath@def{atan}{192}{10.86852} \pgfmath@def{atan}{193}{10.92377} \pgfmath@def{atan}{194}{10.97900} \pgfmath@def{atan}{195}{11.03420} \pgfmath@def{atan}{196}{11.08939} \pgfmath@def{atan}{197}{11.14456} \pgfmath@def{atan}{198}{11.19970} \pgfmath@def{atan}{199}{11.25483} \pgfmath@def{atan}{200}{11.30993} \pgfmath@def{atan}{201}{11.36501} \pgfmath@def{atan}{202}{11.42007} \pgfmath@def{atan}{203}{11.47511} \pgfmath@def{atan}{204}{11.53013} \pgfmath@def{atan}{205}{11.58512} \pgfmath@def{atan}{206}{11.64010} \pgfmath@def{atan}{207}{11.69505} \pgfmath@def{atan}{208}{11.74998} \pgfmath@def{atan}{209}{11.80489} \pgfmath@def{atan}{210}{11.85977} \pgfmath@def{atan}{211}{11.91464} \pgfmath@def{atan}{212}{11.96948} \pgfmath@def{atan}{213}{12.02430} \pgfmath@def{atan}{214}{12.07910} \pgfmath@def{atan}{215}{12.13388} \pgfmath@def{atan}{216}{12.18863} \pgfmath@def{atan}{217}{12.24336} \pgfmath@def{atan}{218}{12.29807} \pgfmath@def{atan}{219}{12.35275} \pgfmath@def{atan}{220}{12.40741} \pgfmath@def{atan}{221}{12.46205} \pgfmath@def{atan}{222}{12.51667} \pgfmath@def{atan}{223}{12.57126} \pgfmath@def{atan}{224}{12.62583} \pgfmath@def{atan}{225}{12.68038} \pgfmath@def{atan}{226}{12.73490} \pgfmath@def{atan}{227}{12.78940} \pgfmath@def{atan}{228}{12.84388} \pgfmath@def{atan}{229}{12.89833} \pgfmath@def{atan}{230}{12.95276} \pgfmath@def{atan}{231}{13.00717} \pgfmath@def{atan}{232}{13.06155} \pgfmath@def{atan}{233}{13.11590} \pgfmath@def{atan}{234}{13.17024} \pgfmath@def{atan}{235}{13.22455} \pgfmath@def{atan}{236}{13.27883} \pgfmath@def{atan}{237}{13.33309} \pgfmath@def{atan}{238}{13.38733} \pgfmath@def{atan}{239}{13.44154} \pgfmath@def{atan}{240}{13.49573} \pgfmath@def{atan}{241}{13.54989} \pgfmath@def{atan}{242}{13.60403} \pgfmath@def{atan}{243}{13.65814} \pgfmath@def{atan}{244}{13.71223} \pgfmath@def{atan}{245}{13.76630} \pgfmath@def{atan}{246}{13.82034} \pgfmath@def{atan}{247}{13.87435} \pgfmath@def{atan}{248}{13.92834} \pgfmath@def{atan}{249}{13.98230} \pgfmath@def{atan}{250}{14.03624} \pgfmath@def{atan}{251}{14.09015} \pgfmath@def{atan}{252}{14.14404} \pgfmath@def{atan}{253}{14.19790} \pgfmath@def{atan}{254}{14.25174} \pgfmath@def{atan}{255}{14.30555} \pgfmath@def{atan}{256}{14.35933} \pgfmath@def{atan}{257}{14.41309} \pgfmath@def{atan}{258}{14.46682} \pgfmath@def{atan}{259}{14.52053} \pgfmath@def{atan}{260}{14.57421} \pgfmath@def{atan}{261}{14.62787} \pgfmath@def{atan}{262}{14.68149} \pgfmath@def{atan}{263}{14.73510} \pgfmath@def{atan}{264}{14.78867} \pgfmath@def{atan}{265}{14.84222} \pgfmath@def{atan}{266}{14.89575} \pgfmath@def{atan}{267}{14.94924} \pgfmath@def{atan}{268}{15.00271} \pgfmath@def{atan}{269}{15.05615} \pgfmath@def{atan}{270}{15.10957} \pgfmath@def{atan}{271}{15.16296} \pgfmath@def{atan}{272}{15.21632} \pgfmath@def{atan}{273}{15.26966} \pgfmath@def{atan}{274}{15.32297} \pgfmath@def{atan}{275}{15.37625} \pgfmath@def{atan}{276}{15.42950} \pgfmath@def{atan}{277}{15.48273} \pgfmath@def{atan}{278}{15.53593} \pgfmath@def{atan}{279}{15.58910} \pgfmath@def{atan}{280}{15.64224} \pgfmath@def{atan}{281}{15.69536} \pgfmath@def{atan}{282}{15.74845} \pgfmath@def{atan}{283}{15.80151} \pgfmath@def{atan}{284}{15.85454} \pgfmath@def{atan}{285}{15.90755} \pgfmath@def{atan}{286}{15.96052} \pgfmath@def{atan}{287}{16.01347} \pgfmath@def{atan}{288}{16.06640} \pgfmath@def{atan}{289}{16.11929} \pgfmath@def{atan}{290}{16.17215} \pgfmath@def{atan}{291}{16.22499} \pgfmath@def{atan}{292}{16.27780} \pgfmath@def{atan}{293}{16.33058} \pgfmath@def{atan}{294}{16.38333} \pgfmath@def{atan}{295}{16.43605} \pgfmath@def{atan}{296}{16.48875} \pgfmath@def{atan}{297}{16.54142} 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